Constants of motion for superconducting Josephson arrays

We show that series arrays of N identical overdamped Josephson junctions have extremely degenerate dynamics. In particular, we prove that such arrays have N - 3 constants of motion for all N ⩾ 3. The analysis is based on a coordinate transformation that reduces the governing equations to an ( N - 3)-parameter family of low-dimensional systems. In the weak-coupling limit, the reduced equations can be analyzed completely. Either all solutions approach the synchronous state or they converge to a continuous family of incoherent oscillations, depending on a certain parameter value. At the transitional value of this parameter, the system becomes completely integrable. Then the phase space is foliated by invariant two-dimensional tori, for any N ⩾ 3. The infinite- N limit of the system is an integro-partial differential equation with rigorously low-dimensional dynamics. It supports solitons in the integrable case, and chaotic waves in the non-integrable case.